X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2FLAMBDA-TYPES%2FLambdaDelta-1%2Fpr2%2Fprops.ma;h=74e2bf65ec1dd39edd9d6389aaa7a1e8d58064be;hb=e92710b1d9774a6491122668c8463b8658114610;hp=8b11644d3e3fdaf731d86149bf6bc787cd9b1377;hpb=d0982534aee06a30f91a06d2f3e82834b132a3d3;p=helm.git

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr2/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr2/props.ma
index 8b11644d3..74e2bf65e 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr2/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr2/props.ma
@@ -14,13 +14,13 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "pr2/defs.ma".
+include "LambdaDelta-1/pr2/defs.ma".
 
-include "pr0/props.ma".
+include "LambdaDelta-1/pr0/props.ma".
 
-include "getl/drop.ma".
+include "LambdaDelta-1/getl/drop.ma".
 
-include "getl/clear.ma".
+include "LambdaDelta-1/getl/clear.ma".
 
 theorem pr2_thin_dx:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall 
@@ -62,160 +62,120 @@ theorem pr2_head_2:
 t2)))))))
 \def
  \lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
-(k: K).(K_ind (\lambda (k0: K).((pr2 (CHead c k0 u) t1 t2) \to (pr2 c (THead 
-k0 u t1) (THead k0 u t2)))) (\lambda (b: B).(\lambda (H: (pr2 (CHead c (Bind 
-b) u) t1 t2)).(let H0 \def (match H in pr2 return (\lambda (c0: C).(\lambda 
-(t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 (CHead c (Bind 
-b) u)) \to ((eq T t t1) \to ((eq T t0 t2) \to (pr2 c (THead (Bind b) u t1) 
-(THead (Bind b) u t2))))))))) with [(pr2_free c0 t0 t3 H0) \Rightarrow 
-(\lambda (H1: (eq C c0 (CHead c (Bind b) u))).(\lambda (H2: (eq T t0 
-t1)).(\lambda (H3: (eq T t3 t2)).(eq_ind C (CHead c (Bind b) u) (\lambda (_: 
-C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c (THead (Bind 
-b) u t1) (THead (Bind b) u t2)))))) (\lambda (H4: (eq T t0 t1)).(eq_ind T t1 
-(\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (pr2 c (THead (Bind b) u 
-t1) (THead (Bind b) u t2))))) (\lambda (H5: (eq T t3 t2)).(eq_ind T t2 
-(\lambda (t: T).((pr0 t1 t) \to (pr2 c (THead (Bind b) u t1) (THead (Bind b) 
-u t2)))) (\lambda (H6: (pr0 t1 t2)).(pr2_free c (THead (Bind b) u t1) (THead 
-(Bind b) u t2) (pr0_comp u u (pr0_refl u) t1 t2 H6 (Bind b)))) t3 (sym_eq T 
-t3 t2 H5))) t0 (sym_eq T t0 t1 H4))) c0 (sym_eq C c0 (CHead c (Bind b) u) H1) 
-H2 H3 H0)))) | (pr2_delta c0 d u0 i H0 t0 t3 H1 t H2) \Rightarrow (\lambda 
-(H3: (eq C c0 (CHead c (Bind b) u))).(\lambda (H4: (eq T t0 t1)).(\lambda 
-(H5: (eq T t t2)).(eq_ind C (CHead c (Bind b) u) (\lambda (c1: C).((eq T t0 
-t1) \to ((eq T t t2) \to ((getl i c1 (CHead d (Bind Abbr) u0)) \to ((pr0 t0 
-t3) \to ((subst0 i u0 t3 t) \to (pr2 c (THead (Bind b) u t1) (THead (Bind b) 
-u t2)))))))) (\lambda (H6: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T 
-t t2) \to ((getl i (CHead c (Bind b) u) (CHead d (Bind Abbr) u0)) \to ((pr0 
-t4 t3) \to ((subst0 i u0 t3 t) \to (pr2 c (THead (Bind b) u t1) (THead (Bind 
-b) u t2))))))) (\lambda (H7: (eq T t t2)).(eq_ind T t2 (\lambda (t4: 
-T).((getl i (CHead c (Bind b) u) (CHead d (Bind Abbr) u0)) \to ((pr0 t1 t3) 
-\to ((subst0 i u0 t3 t4) \to (pr2 c (THead (Bind b) u t1) (THead (Bind b) u 
-t2)))))) (\lambda (H8: (getl i (CHead c (Bind b) u) (CHead d (Bind Abbr) 
-u0))).(\lambda (H9: (pr0 t1 t3)).(\lambda (H10: (subst0 i u0 t3 t2)).(nat_ind 
-(\lambda (n: nat).((getl n (CHead c (Bind b) u) (CHead d (Bind Abbr) u0)) \to 
-((subst0 n u0 t3 t2) \to (pr2 c (THead (Bind b) u t1) (THead (Bind b) u 
-t2))))) (\lambda (H11: (getl O (CHead c (Bind b) u) (CHead d (Bind Abbr) 
-u0))).(\lambda (H12: (subst0 O u0 t3 t2)).(let H13 \def (f_equal C C (\lambda 
-(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d 
-| (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u0) (CHead c (Bind b) 
-u) (clear_gen_bind b c (CHead d (Bind Abbr) u0) u (getl_gen_O (CHead c (Bind 
-b) u) (CHead d (Bind Abbr) u0) H11))) in ((let H14 \def (f_equal C B (\lambda 
-(e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow 
-Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K return (\lambda (_: K).B) 
-with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d 
+(k: K).(\lambda (H: (pr2 (CHead c k u) t1 t2)).(insert_eq C (CHead c k u) 
+(\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c (THead k u t1) (THead 
+k u t2))) (\lambda (y: C).(\lambda (H0: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: 
+C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c k u)) \to (pr2 c 
+(THead k u t) (THead k u t0)))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda 
+(t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c k 
+u))).(pr2_free c (THead k u t3) (THead k u t4) (pr0_comp u u (pr0_refl u) t3 
+t4 H1 k))))))) (K_ind (\lambda (k0: K).(\forall (c0: C).(\forall (d: 
+C).(\forall (u0: T).(\forall (i: nat).((getl i c0 (CHead d (Bind Abbr) u0)) 
+\to (\forall (t3: T).(\forall (t4: T).((pr0 t3 t4) \to (\forall (t: 
+T).((subst0 i u0 t4 t) \to ((eq C c0 (CHead c k0 u)) \to (pr2 c (THead k0 u 
+t3) (THead k0 u t)))))))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda (d: 
+C).(\lambda (u0: T).(\lambda (i: nat).(nat_ind (\lambda (n: nat).((getl n c0 
+(CHead d (Bind Abbr) u0)) \to (\forall (t3: T).(\forall (t4: T).((pr0 t3 t4) 
+\to (\forall (t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead c (Bind b) u)) 
+\to (pr2 c (THead (Bind b) u t3) (THead (Bind b) u t)))))))))) (\lambda (H1: 
+(getl O c0 (CHead d (Bind Abbr) u0))).(\lambda (t3: T).(\lambda (t4: 
+T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: (subst0 O u0 t4 
+t)).(\lambda (H4: (eq C c0 (CHead c (Bind b) u))).(let H5 \def (eq_ind C c0 
+(\lambda (c1: C).(getl O c1 (CHead d (Bind Abbr) u0))) H1 (CHead c (Bind b) 
+u) H4) in (let H6 \def (f_equal C C (\lambda (e: C).(match e in C return 
+(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow 
+c1])) (CHead d (Bind Abbr) u0) (CHead c (Bind b) u) (clear_gen_bind b c 
+(CHead d (Bind Abbr) u0) u (getl_gen_O (CHead c (Bind b) u) (CHead d (Bind 
+Abbr) u0) H5))) in ((let H7 \def (f_equal C B (\lambda (e: C).(match e in C 
+return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) 
+\Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b0) 
+\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u0) 
+(CHead c (Bind b) u) (clear_gen_bind b c (CHead d (Bind Abbr) u0) u 
+(getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) H5))) in ((let H8 
+\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) 
+with [(CSort _) \Rightarrow u0 | (CHead _ _ t0) \Rightarrow t0])) (CHead d 
 (Bind Abbr) u0) (CHead c (Bind b) u) (clear_gen_bind b c (CHead d (Bind Abbr) 
-u0) u (getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) H11))) in 
-((let H15 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: 
-C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t4) \Rightarrow t4])) 
-(CHead d (Bind Abbr) u0) (CHead c (Bind b) u) (clear_gen_bind b c (CHead d 
-(Bind Abbr) u0) u (getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) 
-H11))) in (\lambda (H16: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H18 
-\def (eq_ind T u0 (\lambda (t4: T).(subst0 O t4 t3 t2)) H12 u H15) in (eq_ind 
-B Abbr (\lambda (b0: B).(pr2 c (THead (Bind b0) u t1) (THead (Bind b0) u 
-t2))) (pr2_free c (THead (Bind Abbr) u t1) (THead (Bind Abbr) u t2) 
-(pr0_delta u u (pr0_refl u) t1 t3 H9 t2 H18)) b H16))))) H14)) H13)))) 
-(\lambda (i0: nat).(\lambda (_: (((getl i0 (CHead c (Bind b) u) (CHead d 
-(Bind Abbr) u0)) \to ((subst0 i0 u0 t3 t2) \to (pr2 c (THead (Bind b) u t1) 
-(THead (Bind b) u t2)))))).(\lambda (H11: (getl (S i0) (CHead c (Bind b) u) 
-(CHead d (Bind Abbr) u0))).(\lambda (H12: (subst0 (S i0) u0 t3 
-t2)).(pr2_delta c d u0 (r (Bind b) i0) (getl_gen_S (Bind b) c (CHead d (Bind 
-Abbr) u0) u i0 H11) (THead (Bind b) u t1) (THead (Bind b) u t3) (pr0_comp u u 
-(pr0_refl u) t1 t3 H9 (Bind b)) (THead (Bind b) u t2) (subst0_snd (Bind b) u0 
-t2 t3 (r (Bind b) i0) H12 u)))))) i H8 H10)))) t (sym_eq T t t2 H7))) t0 
-(sym_eq T t0 t1 H6))) c0 (sym_eq C c0 (CHead c (Bind b) u) H3) H4 H5 H0 H1 
-H2))))]) in (H0 (refl_equal C (CHead c (Bind b) u)) (refl_equal T t1) 
-(refl_equal T t2))))) (\lambda (f: F).(\lambda (H: (pr2 (CHead c (Flat f) u) 
-t1 t2)).(let H0 \def (match H in pr2 return (\lambda (c0: C).(\lambda (t: 
-T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 (CHead c (Flat f) 
-u)) \to ((eq T t t1) \to ((eq T t0 t2) \to (pr2 c (THead (Flat f) u t1) 
-(THead (Flat f) u t2))))))))) with [(pr2_free c0 t0 t3 H0) \Rightarrow 
-(\lambda (H1: (eq C c0 (CHead c (Flat f) u))).(\lambda (H2: (eq T t0 
-t1)).(\lambda (H3: (eq T t3 t2)).(eq_ind C (CHead c (Flat f) u) (\lambda (_: 
-C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c (THead (Flat 
-f) u t1) (THead (Flat f) u t2)))))) (\lambda (H4: (eq T t0 t1)).(eq_ind T t1 
-(\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (pr2 c (THead (Flat f) u 
-t1) (THead (Flat f) u t2))))) (\lambda (H5: (eq T t3 t2)).(eq_ind T t2 
-(\lambda (t: T).((pr0 t1 t) \to (pr2 c (THead (Flat f) u t1) (THead (Flat f) 
-u t2)))) (\lambda (H6: (pr0 t1 t2)).(pr2_free c (THead (Flat f) u t1) (THead 
-(Flat f) u t2) (pr0_comp u u (pr0_refl u) t1 t2 H6 (Flat f)))) t3 (sym_eq T 
-t3 t2 H5))) t0 (sym_eq T t0 t1 H4))) c0 (sym_eq C c0 (CHead c (Flat f) u) H1) 
-H2 H3 H0)))) | (pr2_delta c0 d u0 i H0 t0 t3 H1 t H2) \Rightarrow (\lambda 
-(H3: (eq C c0 (CHead c (Flat f) u))).(\lambda (H4: (eq T t0 t1)).(\lambda 
-(H5: (eq T t t2)).(eq_ind C (CHead c (Flat f) u) (\lambda (c1: C).((eq T t0 
-t1) \to ((eq T t t2) \to ((getl i c1 (CHead d (Bind Abbr) u0)) \to ((pr0 t0 
-t3) \to ((subst0 i u0 t3 t) \to (pr2 c (THead (Flat f) u t1) (THead (Flat f) 
-u t2)))))))) (\lambda (H6: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T 
-t t2) \to ((getl i (CHead c (Flat f) u) (CHead d (Bind Abbr) u0)) \to ((pr0 
-t4 t3) \to ((subst0 i u0 t3 t) \to (pr2 c (THead (Flat f) u t1) (THead (Flat 
-f) u t2))))))) (\lambda (H7: (eq T t t2)).(eq_ind T t2 (\lambda (t4: 
-T).((getl i (CHead c (Flat f) u) (CHead d (Bind Abbr) u0)) \to ((pr0 t1 t3) 
-\to ((subst0 i u0 t3 t4) \to (pr2 c (THead (Flat f) u t1) (THead (Flat f) u 
-t2)))))) (\lambda (H8: (getl i (CHead c (Flat f) u) (CHead d (Bind Abbr) 
-u0))).(\lambda (H9: (pr0 t1 t3)).(\lambda (H10: (subst0 i u0 t3 t2)).(nat_ind 
-(\lambda (n: nat).((getl n (CHead c (Flat f) u) (CHead d (Bind Abbr) u0)) \to 
-((subst0 n u0 t3 t2) \to (pr2 c (THead (Flat f) u t1) (THead (Flat f) u 
-t2))))) (\lambda (H11: (getl O (CHead c (Flat f) u) (CHead d (Bind Abbr) 
-u0))).(\lambda (H12: (subst0 O u0 t3 t2)).(pr2_delta c d u0 O (getl_intro O c 
-(CHead d (Bind Abbr) u0) c (drop_refl c) (clear_gen_flat f c (CHead d (Bind 
-Abbr) u0) u (getl_gen_O (CHead c (Flat f) u) (CHead d (Bind Abbr) u0) H11))) 
-(THead (Flat f) u t1) (THead (Flat f) u t3) (pr0_comp u u (pr0_refl u) t1 t3 
-H9 (Flat f)) (THead (Flat f) u t2) (subst0_snd (Flat f) u0 t2 t3 O H12 u)))) 
-(\lambda (i0: nat).(\lambda (_: (((getl i0 (CHead c (Flat f) u) (CHead d 
-(Bind Abbr) u0)) \to ((subst0 i0 u0 t3 t2) \to (pr2 c (THead (Flat f) u t1) 
-(THead (Flat f) u t2)))))).(\lambda (H11: (getl (S i0) (CHead c (Flat f) u) 
-(CHead d (Bind Abbr) u0))).(\lambda (H12: (subst0 (S i0) u0 t3 
-t2)).(pr2_delta c d u0 (r (Flat f) i0) (getl_gen_S (Flat f) c (CHead d (Bind 
-Abbr) u0) u i0 H11) (THead (Flat f) u t1) (THead (Flat f) u t3) (pr0_comp u u 
-(pr0_refl u) t1 t3 H9 (Flat f)) (THead (Flat f) u t2) (subst0_snd (Flat f) u0 
-t2 t3 (r (Flat f) i0) H12 u)))))) i H8 H10)))) t (sym_eq T t t2 H7))) t0 
-(sym_eq T t0 t1 H6))) c0 (sym_eq C c0 (CHead c (Flat f) u) H3) H4 H5 H0 H1 
-H2))))]) in (H0 (refl_equal C (CHead c (Flat f) u)) (refl_equal T t1) 
-(refl_equal T t2))))) k))))).
+u0) u (getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) H5))) in 
+(\lambda (H9: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H11 \def (eq_ind T 
+u0 (\lambda (t0: T).(subst0 O t0 t4 t)) H3 u H8) in (eq_ind B Abbr (\lambda 
+(b0: B).(pr2 c (THead (Bind b0) u t3) (THead (Bind b0) u t))) (pr2_free c 
+(THead (Bind Abbr) u t3) (THead (Bind Abbr) u t) (pr0_delta u u (pr0_refl u) 
+t3 t4 H2 t H11)) b H9))))) H7)) H6)))))))))) (\lambda (n: nat).(\lambda (H1: 
+(((getl n c0 (CHead d (Bind Abbr) u0)) \to (\forall (t3: T).(\forall (t4: 
+T).((pr0 t3 t4) \to (\forall (t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead 
+c (Bind b) u)) \to (pr2 c (THead (Bind b) u t3) (THead (Bind b) u 
+t))))))))))).(\lambda (H2: (getl (S n) c0 (CHead d (Bind Abbr) u0))).(\lambda 
+(t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda (t: T).(\lambda 
+(H4: (subst0 (S n) u0 t4 t)).(\lambda (H5: (eq C c0 (CHead c (Bind b) 
+u))).(let H6 \def (eq_ind C c0 (\lambda (c1: C).(getl (S n) c1 (CHead d (Bind 
+Abbr) u0))) H2 (CHead c (Bind b) u) H5) in (let H7 \def (eq_ind C c0 (\lambda 
+(c1: C).((getl n c1 (CHead d (Bind Abbr) u0)) \to (\forall (t5: T).(\forall 
+(t6: T).((pr0 t5 t6) \to (\forall (t0: T).((subst0 n u0 t6 t0) \to ((eq C c1 
+(CHead c (Bind b) u)) \to (pr2 c (THead (Bind b) u t5) (THead (Bind b) u 
+t0)))))))))) H1 (CHead c (Bind b) u) H5) in (pr2_delta c d u0 (r (Bind b) n) 
+(getl_gen_S (Bind b) c (CHead d (Bind Abbr) u0) u n H6) (THead (Bind b) u t3) 
+(THead (Bind b) u t4) (pr0_comp u u (pr0_refl u) t3 t4 H3 (Bind b)) (THead 
+(Bind b) u t) (subst0_snd (Bind b) u0 t t4 (r (Bind b) n) H4 u))))))))))))) 
+i)))))) (\lambda (f: F).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: 
+T).(\lambda (i: nat).(nat_ind (\lambda (n: nat).((getl n c0 (CHead d (Bind 
+Abbr) u0)) \to (\forall (t3: T).(\forall (t4: T).((pr0 t3 t4) \to (\forall 
+(t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead c (Flat f) u)) \to (pr2 c 
+(THead (Flat f) u t3) (THead (Flat f) u t)))))))))) (\lambda (H1: (getl O c0 
+(CHead d (Bind Abbr) u0))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: 
+(pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: (subst0 O u0 t4 t)).(\lambda (H4: 
+(eq C c0 (CHead c (Flat f) u))).(let H5 \def (eq_ind C c0 (\lambda (c1: 
+C).(getl O c1 (CHead d (Bind Abbr) u0))) H1 (CHead c (Flat f) u) H4) in 
+(pr2_delta c d u0 O (getl_intro O c (CHead d (Bind Abbr) u0) c (drop_refl c) 
+(clear_gen_flat f c (CHead d (Bind Abbr) u0) u (getl_gen_O (CHead c (Flat f) 
+u) (CHead d (Bind Abbr) u0) H5))) (THead (Flat f) u t3) (THead (Flat f) u t4) 
+(pr0_comp u u (pr0_refl u) t3 t4 H2 (Flat f)) (THead (Flat f) u t) 
+(subst0_snd (Flat f) u0 t t4 O H3 u)))))))))) (\lambda (n: nat).(\lambda (H1: 
+(((getl n c0 (CHead d (Bind Abbr) u0)) \to (\forall (t3: T).(\forall (t4: 
+T).((pr0 t3 t4) \to (\forall (t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead 
+c (Flat f) u)) \to (pr2 c (THead (Flat f) u t3) (THead (Flat f) u 
+t))))))))))).(\lambda (H2: (getl (S n) c0 (CHead d (Bind Abbr) u0))).(\lambda 
+(t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda (t: T).(\lambda 
+(H4: (subst0 (S n) u0 t4 t)).(\lambda (H5: (eq C c0 (CHead c (Flat f) 
+u))).(let H6 \def (eq_ind C c0 (\lambda (c1: C).(getl (S n) c1 (CHead d (Bind 
+Abbr) u0))) H2 (CHead c (Flat f) u) H5) in (let H7 \def (eq_ind C c0 (\lambda 
+(c1: C).((getl n c1 (CHead d (Bind Abbr) u0)) \to (\forall (t5: T).(\forall 
+(t6: T).((pr0 t5 t6) \to (\forall (t0: T).((subst0 n u0 t6 t0) \to ((eq C c1 
+(CHead c (Flat f) u)) \to (pr2 c (THead (Flat f) u t5) (THead (Flat f) u 
+t0)))))))))) H1 (CHead c (Flat f) u) H5) in (pr2_delta c d u0 (r (Flat f) n) 
+(getl_gen_S (Flat f) c (CHead d (Bind Abbr) u0) u n H6) (THead (Flat f) u t3) 
+(THead (Flat f) u t4) (pr0_comp u u (pr0_refl u) t3 t4 H3 (Flat f)) (THead 
+(Flat f) u t) (subst0_snd (Flat f) u0 t t4 (r (Flat f) n) H4 u))))))))))))) 
+i)))))) k) y t1 t2 H0))) H)))))).
 
 theorem clear_pr2_trans:
  \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pr2 c2 t1 t2) \to 
 (\forall (c1: C).((clear c1 c2) \to (pr2 c1 t1 t2))))))
 \def
  \lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c2 t1 
-t2)).(\lambda (c1: C).(\lambda (H0: (clear c1 c2)).(let H1 \def (match H in 
-pr2 return (\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 
-c t t0)).((eq C c c2) \to ((eq T t t1) \to ((eq T t0 t2) \to (pr2 c1 t1 
-t2)))))))) with [(pr2_free c t0 t3 H1) \Rightarrow (\lambda (H2: (eq C c 
-c2)).(\lambda (H3: (eq T t0 t1)).(\lambda (H4: (eq T t3 t2)).(eq_ind C c2 
-(\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c1 
-t1 t2))))) (\lambda (H5: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 
-t2) \to ((pr0 t t3) \to (pr2 c1 t1 t2)))) (\lambda (H6: (eq T t3 t2)).(eq_ind 
-T t2 (\lambda (t: T).((pr0 t1 t) \to (pr2 c1 t1 t2))) (\lambda (H7: (pr0 t1 
-t2)).(pr2_free c1 t1 t2 H7)) t3 (sym_eq T t3 t2 H6))) t0 (sym_eq T t0 t1 
-H5))) c (sym_eq C c c2 H2) H3 H4 H1)))) | (pr2_delta c d u i H1 t0 t3 H2 t 
-H3) \Rightarrow (\lambda (H4: (eq C c c2)).(\lambda (H5: (eq T t0 
-t1)).(\lambda (H6: (eq T t t2)).(eq_ind C c2 (\lambda (c0: C).((eq T t0 t1) 
-\to ((eq T t t2) \to ((getl i c0 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t3) 
-\to ((subst0 i u t3 t) \to (pr2 c1 t1 t2))))))) (\lambda (H7: (eq T t0 
-t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i c2 (CHead d 
-(Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pr2 c1 t1 
-t2)))))) (\lambda (H8: (eq T t t2)).(eq_ind T t2 (\lambda (t4: T).((getl i c2 
-(CHead d (Bind Abbr) u)) \to ((pr0 t1 t3) \to ((subst0 i u t3 t4) \to (pr2 c1 
-t1 t2))))) (\lambda (H9: (getl i c2 (CHead d (Bind Abbr) u))).(\lambda (H10: 
-(pr0 t1 t3)).(\lambda (H11: (subst0 i u t3 t2)).(pr2_delta c1 d u i 
-(clear_getl_trans i c2 (CHead d (Bind Abbr) u) H9 c1 H0) t1 t3 H10 t2 H11)))) 
-t (sym_eq T t t2 H8))) t0 (sym_eq T t0 t1 H7))) c (sym_eq C c c2 H4) H5 H6 H1 
-H2 H3))))]) in (H1 (refl_equal C c2) (refl_equal T t1) (refl_equal T 
-t2)))))))).
+t2)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).(\forall (c1: 
+C).((clear c1 c) \to (pr2 c1 t t0)))))) (\lambda (c: C).(\lambda (t3: 
+T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda (c1: C).(\lambda (_: 
+(clear c1 c)).(pr2_free c1 t3 t4 H0))))))) (\lambda (c: C).(\lambda (d: 
+C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind 
+Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 
+t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (c1: 
+C).(\lambda (H3: (clear c1 c)).(pr2_delta c1 d u i (clear_getl_trans i c 
+(CHead d (Bind Abbr) u) H0 c1 H3) t3 t4 H1 t H2))))))))))))) c2 t1 t2 H)))).
 
 theorem pr2_cflat:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall 
 (f: F).(\forall (v: T).(pr2 (CHead c (Flat f) v) t1 t2))))))
 \def
  \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 
-t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (f: 
-F).(\forall (v: T).(pr2 (CHead c0 (Flat f) v) t t0)))))) (\lambda (c0: 
-C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda (f: 
-F).(\lambda (v: T).(pr2_free (CHead c0 (Flat f) v) t3 t4 H0))))))) (\lambda 
-(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl 
-i c0 (CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda 
-(H1: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda 
-(f: F).(\lambda (v: T).(pr2_delta (CHead c0 (Flat f) v) d u i (getl_flat c0 
-(CHead d (Bind Abbr) u) i H0 f v) t3 t4 H1 t H2))))))))))))) c t1 t2 H)))).
+t2)).(\lambda (f: F).(\lambda (v: T).(pr2_ind (\lambda (c0: C).(\lambda (t: 
+T).(\lambda (t0: T).(pr2 (CHead c0 (Flat f) v) t t0)))) (\lambda (c0: 
+C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(pr2_free 
+(CHead c0 (Flat f) v) t3 t4 H0))))) (\lambda (c0: C).(\lambda (d: C).(\lambda 
+(u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) 
+u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda 
+(t: T).(\lambda (H2: (subst0 i u t4 t)).(pr2_delta (CHead c0 (Flat f) v) d u 
+i (getl_flat c0 (CHead d (Bind Abbr) u) i H0 f v) t3 t4 H1 t H2))))))))))) c 
+t1 t2 H)))))).
 
 theorem pr2_ctail:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall 
@@ -288,50 +248,36 @@ h d t1) (lift h d t2)))))))))
 \def
  \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda 
 (H: (drop h d c e)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr2 e t1 
-t2)).(let H1 \def (match H0 in pr2 return (\lambda (c0: C).(\lambda (t: 
-T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 e) \to ((eq T t t1) 
-\to ((eq T t0 t2) \to (pr2 c (lift h d t1) (lift h d t2))))))))) with 
-[(pr2_free c0 t0 t3 H1) \Rightarrow (\lambda (H2: (eq C c0 e)).(\lambda (H3: 
-(eq T t0 t1)).(\lambda (H4: (eq T t3 t2)).(eq_ind C e (\lambda (_: C).((eq T 
-t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c (lift h d t1) (lift h d 
-t2)))))) (\lambda (H5: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 
-t2) \to ((pr0 t t3) \to (pr2 c (lift h d t1) (lift h d t2))))) (\lambda (H6: 
-(eq T t3 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t1 t) \to (pr2 c (lift h d 
-t1) (lift h d t2)))) (\lambda (H7: (pr0 t1 t2)).(pr2_free c (lift h d t1) 
-(lift h d t2) (pr0_lift t1 t2 H7 h d))) t3 (sym_eq T t3 t2 H6))) t0 (sym_eq T 
-t0 t1 H5))) c0 (sym_eq C c0 e H2) H3 H4 H1)))) | (pr2_delta c0 d0 u i H1 t0 
-t3 H2 t H3) \Rightarrow (\lambda (H4: (eq C c0 e)).(\lambda (H5: (eq T t0 
-t1)).(\lambda (H6: (eq T t t2)).(eq_ind C e (\lambda (c1: C).((eq T t0 t1) 
-\to ((eq T t t2) \to ((getl i c1 (CHead d0 (Bind Abbr) u)) \to ((pr0 t0 t3) 
-\to ((subst0 i u t3 t) \to (pr2 c (lift h d t1) (lift h d t2)))))))) (\lambda 
-(H7: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i e 
-(CHead d0 (Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pr2 c 
-(lift h d t1) (lift h d t2))))))) (\lambda (H8: (eq T t t2)).(eq_ind T t2 
-(\lambda (t4: T).((getl i e (CHead d0 (Bind Abbr) u)) \to ((pr0 t1 t3) \to 
-((subst0 i u t3 t4) \to (pr2 c (lift h d t1) (lift h d t2)))))) (\lambda (H9: 
-(getl i e (CHead d0 (Bind Abbr) u))).(\lambda (H10: (pr0 t1 t3)).(\lambda 
-(H11: (subst0 i u t3 t2)).(lt_le_e i d (pr2 c (lift h d t1) (lift h d t2)) 
-(\lambda (H12: (lt i d)).(let H13 \def (drop_getl_trans_le i d (le_S_n i d 
-(le_S (S i) d H12)) c e h H (CHead d0 (Bind Abbr) u) H9) in (ex3_2_ind C C 
-(\lambda (e0: C).(\lambda (_: C).(drop i O c e0))) (\lambda (e0: C).(\lambda 
-(e1: C).(drop h (minus d i) e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear 
-e1 (CHead d0 (Bind Abbr) u)))) (pr2 c (lift h d t1) (lift h d t2)) (\lambda 
-(x0: C).(\lambda (x1: C).(\lambda (H14: (drop i O c x0)).(\lambda (H15: (drop 
-h (minus d i) x0 x1)).(\lambda (H16: (clear x1 (CHead d0 (Bind Abbr) 
-u))).(let H17 \def (eq_ind nat (minus d i) (\lambda (n: nat).(drop h n x0 
-x1)) H15 (S (minus d (S i))) (minus_x_Sy d i H12)) in (let H18 \def 
-(drop_clear_S x1 x0 h (minus d (S i)) H17 Abbr d0 u H16) in (ex2_ind C 
-(\lambda (c1: C).(clear x0 (CHead c1 (Bind Abbr) (lift h (minus d (S i)) 
-u)))) (\lambda (c1: C).(drop h (minus d (S i)) c1 d0)) (pr2 c (lift h d t1) 
-(lift h d t2)) (\lambda (x: C).(\lambda (H19: (clear x0 (CHead x (Bind Abbr) 
-(lift h (minus d (S i)) u)))).(\lambda (_: (drop h (minus d (S i)) x 
-d0)).(pr2_delta c x (lift h (minus d (S i)) u) i (getl_intro i c (CHead x 
-(Bind Abbr) (lift h (minus d (S i)) u)) x0 H14 H19) (lift h d t1) (lift h d 
-t3) (pr0_lift t1 t3 H10 h d) (lift h d t2) (subst0_lift_lt t3 t2 u i H11 d 
-H12 h))))) H18)))))))) H13))) (\lambda (H12: (le d i)).(pr2_delta c d0 u 
-(plus i h) (drop_getl_trans_ge i c e d h H (CHead d0 (Bind Abbr) u) H9 H12) 
-(lift h d t1) (lift h d t3) (pr0_lift t1 t3 H10 h d) (lift h d t2) 
-(subst0_lift_ge t3 t2 u i h H11 d H12))))))) t (sym_eq T t t2 H8))) t0 
-(sym_eq T t0 t1 H7))) c0 (sym_eq C c0 e H4) H5 H6 H1 H2 H3))))]) in (H1 
-(refl_equal C e) (refl_equal T t1) (refl_equal T t2)))))))))).
+t2)).(insert_eq C e (\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c 
+(lift h d t1) (lift h d t2))) (\lambda (y: C).(\lambda (H1: (pr2 y t1 
+t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 e) 
+\to (pr2 c (lift h d t) (lift h d t0)))))) (\lambda (c0: C).(\lambda (t3: 
+T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (_: (eq C c0 
+e)).(pr2_free c (lift h d t3) (lift h d t4) (pr0_lift t3 t4 H2 h d))))))) 
+(\lambda (c0: C).(\lambda (d0: C).(\lambda (u: T).(\lambda (i: nat).(\lambda 
+(H2: (getl i c0 (CHead d0 (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: 
+T).(\lambda (H3: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H4: (subst0 i u t4 
+t)).(\lambda (H5: (eq C c0 e)).(let H6 \def (eq_ind C c0 (\lambda (c1: 
+C).(getl i c1 (CHead d0 (Bind Abbr) u))) H2 e H5) in (lt_le_e i d (pr2 c 
+(lift h d t3) (lift h d t)) (\lambda (H7: (lt i d)).(let H8 \def 
+(drop_getl_trans_le i d (le_S_n i d (le_S (S i) d H7)) c e h H (CHead d0 
+(Bind Abbr) u) H6) in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i 
+O c e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i) e0 e1))) 
+(\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d0 (Bind Abbr) u)))) (pr2 c 
+(lift h d t3) (lift h d t)) (\lambda (x0: C).(\lambda (x1: C).(\lambda (H9: 
+(drop i O c x0)).(\lambda (H10: (drop h (minus d i) x0 x1)).(\lambda (H11: 
+(clear x1 (CHead d0 (Bind Abbr) u))).(let H12 \def (eq_ind nat (minus d i) 
+(\lambda (n: nat).(drop h n x0 x1)) H10 (S (minus d (S i))) (minus_x_Sy d i 
+H7)) in (let H13 \def (drop_clear_S x1 x0 h (minus d (S i)) H12 Abbr d0 u 
+H11) in (ex2_ind C (\lambda (c1: C).(clear x0 (CHead c1 (Bind Abbr) (lift h 
+(minus d (S i)) u)))) (\lambda (c1: C).(drop h (minus d (S i)) c1 d0)) (pr2 c 
+(lift h d t3) (lift h d t)) (\lambda (x: C).(\lambda (H14: (clear x0 (CHead x 
+(Bind Abbr) (lift h (minus d (S i)) u)))).(\lambda (_: (drop h (minus d (S 
+i)) x d0)).(pr2_delta c x (lift h (minus d (S i)) u) i (getl_intro i c (CHead 
+x (Bind Abbr) (lift h (minus d (S i)) u)) x0 H9 H14) (lift h d t3) (lift h d 
+t4) (pr0_lift t3 t4 H3 h d) (lift h d t) (subst0_lift_lt t4 t u i H4 d H7 
+h))))) H13)))))))) H8))) (\lambda (H7: (le d i)).(pr2_delta c d0 u (plus i h) 
+(drop_getl_trans_ge i c e d h H (CHead d0 (Bind Abbr) u) H6 H7) (lift h d t3) 
+(lift h d t4) (pr0_lift t3 t4 H3 h d) (lift h d t) (subst0_lift_ge t4 t u i h 
+H4 d H7)))))))))))))))) y t1 t2 H1))) H0)))))))).